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Surely the inverse Galois problem is known over $\mathbf{C}(x)$: The Galois group of the maximal extension of $\mathbf{C}(x)$ unramified away from $n+1$ given primes of $\mathbf{C}[x]$ is the free profinite group on $n$ generators. Any finite group $G$ is a quotient of such a group, so there exists a finite Galois extension $L/\mathbf{C}(x)$ with Galois group $G$.

Then $L\otimes_{\mathbf{C}(x)}\mathbf{C}(x,y)$ is a finite Galois extension of $\mathbf{C}(x,y)$ with Galois group $G$.